View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNALOF CCXAPUlAlIONAl.AND APPLIED MATHEMATICS Journal of Computational and Applied Mathematics 94 (1998) 55-67 A numerical algorithm for the solution of Signorini problems J.M. Aitchison*, M.W. Poole Applied Mathematics and Operational Research Group, Cranjeld University, RMCS Shrivenham, Swindon, SN6 8LA, UK Received 2 October 1996; received in revised form 3 December 1997 Abstract In this paper we propose a new iterative algorithm for the solution of a certain class of Signorini problems. Such problems arise in the modelling of a variety of physical phenomena and usually involve the determination of an unknown free boundary. Here we describe a way of locating the free boundary directly and provide a proof that the algorithm converges when used with analytic methods. The advantage of this algorithm is that it can be used in conjunction with any numerical method with minimal development of extra code. We demonstrate its application with the boundary element method to some physical problems in both two and three dimensions. @ 1998 Elsevier Science B.V. All rights reserved. AMS classification: 65N38, 35R35 Keywords: Signorini; Free boundary problems; Boundary element methods; Complementarity relations 1. Introduction A Signorini-type problem is one in which the boundary conditions for a governing partial dif- ferential equation are in the form of inequalities involving the unknown function and its derivative together with a complementarity relation. Such boundary conditions are often referred to as being unilateral. In 1933, Signorini [ 151 studied a contact problem in elasticity with unilateral boundary conditions from which the term Signorini problem was coined. (For a more recent reference consult Glowinski [6]. ) Not surprisingly, most Signorini problems occur in contact problems where we have two or more objects colliding. Typically, boundary conditions are of the form 24”- s < 0, r, < 0, Z,(U, - s) = 0 * Corresponding author. 0377~0427/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved Z’ZZSO377-0427(98)00030-2 56 J.M. Aitchison, M. W. Poole1 Journal of Computational and Applied Mathematics 94 (1998) 5567 applying on the contact part of the boundary. Here u, is the displacement in the direction of the outward normal to the contact surface and r, is the component of the Cauchy stress tensor in the normal direction. The distance between the contact surface and the obstacle (possibly another moving body) is given by s, which may be a given positive function. The case u,=s and r,<O represents the two objects being in the contact whilst the case u,<s and r,=O occurs when the two objects are not in contact. Such boundary conditions in the context of contact problems are often referred to as nonpenetration conditions. Examples of contact problems can be found in [9, 10, 121. More general Signorini problems are sometimes referred to as thin-obstacle problems [4], where the conditions on the boundary are, for example, where 4 is the unknown function to be determined and where f and g may be given functions of position and are known as thin obstacles. Problems of this type studied in the literature include the electropaint problem ([2], the shallow dam problem [ 11) and some optimal shape design problems considered by Neittaanmaki [ 111 and Haslinger [7]. It is known (for example, see [6]) that Signorini problems such as these can be recast as variational inequalities and hence can be solved directly by the finite element method (FEM). Its ease in dealing with the problems posed by the inequality boundary conditions has made the FEM the most popular numerical method for solving Signorini problems and is the basis of the numerical methods used by all of the authors mentioned above. However, in these problems the primary interest is in some or all of the boundary making them ideal candidates for solution by the boundary element method (BEM). However, methods such as the BEM cannot incorporate the inequality boundary conditions directly and so require some method of dealing with them. As an example, Karageorghis [S] considers the BEM as a solution method for the shallow dam problem and obtains results which agree with those of Aitchison et al. [I], but finds it necessary to impose certain restrictions on the form of the surface profile to obtain a solution. Here we propose an algorithm which handles the boundary conditions in an iterative man- ner. It can be used with the FEM, the BEM or any other numerical method with minimum ex- tra development. The mechanics of the algorithm as applied to a model problem are detailed in Section 2. In Section 3 we present a proof that the iterative algorithm used in conjunction with analytic methods will converge. The implementation of the algorithm in conjunction with numerical methods is then discussed in Section 4. In Section 5 we apply the algorithm implemented using the BEM to problems of electropainting and flow through a shallow dam both of which involve a free boundary. We demonstrate that the free boundary is calculated automatically. J.M. Aitchison, M. W PoolelJournal of Computational and Applied Mathematics 94 (1998) 55-67 57 2. The switching algorithm Consider the solution for the following Signorini problem in an n-dimensional domain Q with boundary iK?o u X& u a&: L[#] = 0 in s2, $=v(x> on aaD, (1) a4 --W(X) on asz, an with the following boundary conditions applying on the remainder of the boundary a&: On aas 4 2 f(x), ad+n 2 g(x), (4 - f)(a+/an - g)=O. (2) Here &xi,xz,. , x,) satisfies the equation in a domain Q in which L is a uniformly elliptic operator with aij = aii(xl,x2,. ,xn), bi = bi(xl, x2,. ,x,) and aij = aii without loss of generality. We recall that the operator L is uniformly elliptic if and only if the operator n a2 ic j=l ai%$%j is elliptic at each point in the domain G?. In matrix notation the ellipticity condition asserts that the symmetric matrix given by the elements aii is positive definite at each point in the domain. The functions f, g, u and w in Eqs. ( 1) and (2) are prescribed functions of space only. The ultimate aim of this paper is to produce a method for the numerical solution of Eqs. ( 1) and (2), but difficulties arise in the imposition of the inequality boundary conditions of Eq. (2). We therefore propose a sequence of problems with straightforward boundary conditions and show that the corresponding sequence of exact solutions to these problems converges to the solution of Eqs. (1) and (2). The modifications which are necessary when the numerical method is used to find an approximate solution of each of the sequence are discussed in Section 4. Assume initially that the boundary condition on X& is c)=~(x> on af2+ (3) Eqs. ( 1) and (3) provide a well-posed problem with solution 4 = #‘)(x1 ,x2,. ,x,). At this point we check the normal derivative of 4(O) on a!& to see if it satisfies the inequality given in Eq. (2). That is, we check to see if a4/an>g(x) on aos. (4) 58 J.M. Aitchison, M. W. Poole/ Journal of Computational and Applied Mathematics 94 (1998) 55-67 For those sections of a!& where inequality (4) is true, the boundary condition (3) is retained for the next iteration. For those sections of a!& where inequality (4) is false, the boundary condition for the next iteration is switched to &p/an = g(x). (5) We now solve Eq. (1) with the appropriate new boundary conditions. This has a well-determined solution 4 = #l)(xr, x2 , . ,x,). We now check to see if the inequalities of Eq. (2) are satisfied, and perform appropriate switches as below: If the boundary condition was 4 = f(x): Check if &j/an >g(x). If true, retain the Dirichlet boundary condition for the next iteration. If false, switch to the Neumann condition given by Eq. (5). If the boundary condition was @/an= g(x): Check if 4>f(x). If true, retain the Neumann boundary condition for the next iteration. If false, switch to the Dirichlet condition given by Eq. (3). The lack of symmetry in the use of inequality signs in the above allows us to deal with the special case where 4 = f(x) and @/art = g(x). If this occurs then the Neumann condition is used in the next iteration. This iterative process continues until none of the switches in boundary conditions described above occurs. The algorithm has then converged. We term this algorithm the switching algorithm due to the way in which it switches boundary conditions at each iteration. 3. Proof of convergence In this section we prove that the switching algorithm will converge when applied to the problem defined by Eqs. (1) and (2) provided that at each stage of the process the resulting well-posed boundary value problem is solved exactly. In practice we will actually use a numerical method to solve each of the sequence of problems. The effects of this are discussed later. To prove convergence, we will need the following theorem. Theorem 1. Let u(x1,x2,.. .,x,) satisfy the equation L[u]&z,aztl+@L, ijzl axiaxj i=, 'axi in u domain Q in which L is u untformly elliptic operator with CZ~=uij(xl,x2,.. .,x,), bi = bi(xl, ~2,.. ,x,,) and aij = aji without 10~s of generality. Suppose that u BM in Sz and that u =h4 at a boundary point p.